B is a multiple of (E) a part of A, and D is a like multiple of (F) a like part of C. This follows from § 5; and hence B: A :: D: C ; (by case III. § 6.)

SCHOLIUM.

Hitherto numbers have been considered as pure or abstract, without any regard to particular kinds and denominations; but § 1, 2, and 3, are also true of numbers of the same particular kinds. For, in § 1, 7 dollars may be represented by 7 dots. In § 2 and 3, 12 crowns are a multiple of 3 crowns, 14 cents are a multiple of 7 cents; 3 crowns are a part of 12 crowns, 7 cents are a part of 14 cents, &c. &c. But a number of one kind cannot be either a multiple or a part of another number of a different kind; for 12 years cannot be a multiple of 3 inches, nor can 7 oxen be a part of 21 sheep.

In § 4, the denominator of a part is a pure number, and can never be of any particular kind whatever. For although 4 men be a third part of 12 men, yet the denominator 3 is a pure number, independent of every particular kind or denomination.

In the rule of multiplication the product is a multiple of the multiplicand, consequently the multiplicand is a part of the product, and the multiplier is the denominator of that part. The product is therefore always of the same kind with the multiplicand, and the multiplier is always a pure number.* Hence a number of one kind cannot be multiplied by a number either of the same or of a different kind: 3 . cannot be multiplied by 57. 7 feet cannot be multiplied by 4 feet; nor can 6 bears be multiplied by 9 asses. All such questions are evidently unscientific and absurd, and serve only to demonstrate the ignorance and stupidity of their authors.

In the rule of division, either a number and one of its parts are given to find the denominator of that part;

See 4, chap. 7, book 1, Malcolm's Arith. page 80 and 134 Fenn's Arith. andart. io of Baron Maferes's Differtation on the ufe of the Negative Sign.

or a number and the denominator of one of its parts are given to find the part. Hence if the divisor and dividend are numbers of the same kind, the quotient must be a pure number; but if the divisor be a pure number, the quotient is of the same kind with the dividend. Let 3 yards be the divisor, and 24 yards the dividend, then the quotient or denominator 8 is a pure number, which indicates that 3 yards are the eighth part of 24 yards. Again, the sixth part of 30 dollars is 5 dollars: here the denominator 6, a pure number, is the divisor, and the quotient 5 dollars is of the same kind with the dividend 30 dollars.

In § 5, one number may be either a multiple or a part of a second number of the same kind; and a third number may be a like multiple or a like part of a fourth, of the same kind, with regard to each other, although of a different kind from that of the two former. Thus 8 yards are a multiple of 2 yards, and 12 dollars are a like multiple of 3 dollars; 2 yards are a part of 8 yards, and 3 dollars are a like part of 12 dollars. The same is also true if the pure numbers 8 and 2 be taken instead of 8 yards and 2 yards.

From § 6, and what has already been shewn in this scholium, it evidently follows, that in four proportional numbers, the first and second must always be of the same kind; and the third and fourth also of one kind, notwithstanding the two latter may be of a kind different from that of the two former. And hence the ridiculous absurdity of the customary manner of stating questions in the rule of proportion, by making the first number of one kind and the second of a different kind; is abundantly manifest.

§ 7. In four proportional numbers, if the first be greater than the second, the third is greater than the fourth; but if the first be less than the second, the third is less than the fourth.

This is evidently true, when four proportional numbers have one of the properties described in the first

B is a multiple of (E) a part of A, and D is a like multiple of (F) a like part of C. This follows from § 5; and hence B: A:: D: C; (by case III. § 6.)

SCHOLIUM.

Hitherto numbers have been considered as pure or abstract, without any regard to particular kinds and denominations; but § 1, 2, and 3, are also true of numbers of the same particular kinds. For, in § 1, 7 dollars may be represented by 7 dots. In § 2 and 3, 12 crowns are a multiple of 3 crowns, 14 cents are a multiple of 7 cents; 3 crowns are a part of 12 crowns, 7 cents are a part of 14 cents, &c. &c. But a number of one kind cannot be either a multiple or a part of another number of a different kind; for 12 years cannot be a multiple of 3 inches, nor can 7 oxen be a part of 21 sheep.

In § 4, the denominator of a part is a pure number, and can never be of any particular kind whatever. For although 4 men be a third part of 12 men, yet the denominator 3 is a pure number, independent of every particular kind or denomination.

In the rule of multiplication the product is a multiple of the multiplicand, consequently the multiplicand is a part of the product, and the multiplier is the denominator of that part. The product is therefore always of the same kind with the multiplicand, and the multiplier is always a pure number. Hence a number of one kind cannot be multiplied by a number either of the same or of a different kind: 3 l. cannot be multiplied by 57. 7 feet cannot be multiplied by 4 feet; nor can 6 bears be multiplied by 9 asses. All such questions are evidently unscientific and absurd, and serve only demonstrate the ignorance and stupidity of their

the rule of division, either a number and one of rts are given to find the denominator of that part;

44, chap. 7, book 1, Malcolm's Arith. page 80 and 134 Fenn's Arith. 10 of Baron Maferes's Differtation on the ufe of the Negative Sign.

or a number and the denominator of one of its parts are given to find the part. Hence if the divisor and dividend are numbers of the same kind, the quotient must be a pure number; but if the divisor be a pure number, the quotient is of the same kind with the dividend. Let 3 yards be the divisor, and 24 yards the dividend, then the quotient or denominator 8 is a pure number, which indicates that 3 yards are the eighth part of 24 yards. Again, the sixth part of 30 dollars is 5 dollars: here the denominator 6, a pure number, is the divisor, and the quotient 5 dollars is of the same kind with the dividend 30 dollars.

In § 5, one number may be either a multiple or a part of a second number of the same kind; and a third number may be a like multiple or a like part of a fourth, of the same kind, with regard to each other, although of a different kind from that of the two former. Thus 8 yards are a multiple of 2 yards, and 12 dollars are a like multiple of 3 dollars; 2 yards are a part of 8 yards, and 3 dollars are a like part of 12 dollars. The same is also true if the pure numbers 8 and 2 be taken instead of 8 yards and 2 yards.

From § 6, and what has already been shewn in this scholium, it evidently follows, that in four proportional numbers, the first and second must always be of the same kind; and the third and fourth also of one kind, notwithstanding the two latter may be of a kind different from that of the two former. And hence the ridiculous absurdity of the customary manner of stating questions in the rule of proportion, by making the first number of one kind and the second of a different kind; is abundantly manifest.

§ 7. In four proportional numbers, if the first be greater than the second, the third is greater than the fourth; but if the first be less than the second, the third is less than the fourth.

This is evidently true, when four proportional numbers have one of the properties described in the first

and second cases of § 6. For any multiple of a number must be greater, and any part must be less, than the number itself. It only remains therefore to prove that the same is also true when four proportional numbers have the properties described in case III. § 6.

A

C

D

E

F..

number, A, be greater shall the third, C, be

Let A, B, C, and D, represent any four pro- B portional numbers, in which A is a multiple of E, a part of B; and C is a like multiple of F, a like part of D. Let the first than the second, B; then greater than the fourth, D. For since E is a part of B, and F is a like part of D; B contains E exactly as often as D contains F (by § 5). And since A is a multiple of E, and C is a like multiple of F; A con-. tains E exactly as often as C contains F (by § 5). But since A is greater than B, A must contain E oftener than B contains E; and consequently C must contain F oftener than D contains F; and therefore C is greater than D.

In the same manner it may be proved that when B is greater than A, D is greater than C; and consequently when A is less than B, C is less than D.

COROLLARY.

If the first of four proportional numbers be equal to the second; the third is equal to the fourth. This naturally follows from the demonstration of this section.

§ 8. In four numbers, proportional according to case III. § 6, if a part of the first, added to the first, be equal to the second; a like part of the third, added to the third, will be equal to the fourth.